Open Metric Ball¶

Definition¶

For a metric space $(X,d)$ , a point $x_0$ in $X$ , and $r > 0$ $0$ , the open metric ball $B_{(X,d)}(x_0,r)$ in $X$ , centred at $x_0$ with radius $r$ in metric $d$ , is defined as the following set:

B_{(X,d)}(x_0,r) = \{x \in X \colon d(x, x_0) < r\}.

If the context is clear, we'll abbreviate this as $B_d(x_0,r)$ or simply $B(x_0,r).$

A ball is a subset of $X$ selected by the distance condition; the ambient space matters, so a real-line ball and a plane ball have different members.

Interpretation¶

An open metric ball is the set of all points whose distance from the centre $x_0$ is strictly less than the radius $r$ . In $\mathbb{R}$ with $d(x,y)=|x-y|$ it is the open interval $(x_0-r,x_0+r)$ $r)$ , and in the plane with the Euclidean distance it is an open disc.

A disc around x0 with radius r, a point inside at distance less than r, and a point on the rim marked as excluded from the open ball.

The green point satisfies $d(\boldsymbol{x},\boldsymbol{x}_0)<r$ and belongs to the ball. The red point on the dashed rim has $d=r$ exactly and is excluded, because the inequality in the definition is strict.

Change the centre, the radius $r$ and the metric: the ball contains the points at distance less than $r$ . Points on the dashed sphere are not included.

The strict inequality is what makes the ball an open set. If $d(x,x_0)<r$ $r$ , put $s=r-d(x,x_0)>0$ $0$ . By the triangle inequality, every $y$ with $d(y,x)<s$ has $d(y,x_0)<s+d(x,x_0)=r$ $r$ , so $B(x,s)\subseteq B(x_0,r)$ $B (x_{0}, r)$ : every point of the ball has a smaller ball around it that still fits inside.

"Ball" refers to the distance condition, not to a round shape. In $\mathbb{R}^2$ the unit ball is a diamond for the taxicab metric and the open square $(-1,1)^2$ for the distance of the maximum norm; here the centre is the origin.

Open balls are the basic building block of analysis in metric spaces: interior points and open sets are defined with them, and convergence $x_i\to x$ means that every ball $B(x,\varepsilon)$ contains all but finitely many terms. On the real line, $B(a,\delta)$ is the set $|x-a|<\delta$ familiar from the absolute value and the $\varepsilon$ - $\delta$ definitions of Math I. Later, continuity between metric spaces is phrased with balls in the same way.

Key Results¶

Open balls are open sets.
A subset of a metric space is open if and only if it can be written as a union of open balls.

Examples¶

In $\mathbb{R}^2$ with the usual distance, $B((0,0),1)$ is the open disc of radius $1$ centred at the origin. Points with $x^2 + y^2 < 1$ are elements of this ball.
In $\mathbb{R}^2$ with the taxicab metric $d_1$ , $B((0,0),1)$ is a diamond shape with vertices at $(\pm1,0)$ and $(0,\pm1)$ .
In any set with the discrete metric, $B(x,1) = \{x\}$ and $B(x,0.5) = \{x\}$ ${x}$ , but $B(x,2)$ is the whole space.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 5

Aliases: open ball, ε-neighbourhood, ε-ball Symbols: delta, element of, epsilon, real numbers, subset

Last updated: 20 October 2026 03:02